How to count the zeros in interval of real or complex function in pari?

[Georgi Guninski <guninski@guninski.com>]

I would like to count the zeros in interval of real or complex function
in pari or some other software.

As far as I can tell this is related to the argument
principle and possibly to |intcirc()|.

The functions I am interested in are:

real:
P(x)=sin(Pi*x)^2+sin(Pi*(gamma(x)+1) /x)^2

complex:
Pc(x)=abs(sin(Pi*x))+abs(sin(Pi*(gamma(x)+1) /x))

The zeros with Re(x) > 1 are exactly the primes.

Thanks.